To investigate the fractional Hermite–Hadamard-type inequalities, a class of the multiplicative fractional integrals having exponential kernels is introduced. Some estimations of upper bounds for the newly introduced class of integral operators are obtained in terms of the established differentiable identity. And our results presented in this study are substantial generalizations of previous findings given by Ali et al. (Asian Res J Math 12:1–11, 2019). Three examples are also provided to identify the correctness of the results that occur with the change of the parameter .
The power generalized Weibull distribution has been proposed recently by [
This paper considers the problem of numerically evaluating discrete barrier option prices when the underlying asset follows the jump-diffusion model with stochastic volatility and stochastic intensity. We derive the three-dimensional characteristic function of the log-asset price, the volatility and the jump intensity. We also provide the approximate formula of the discrete barrier option prices by the three-dimensional Fourier cosine series expansion (3D-COS) method. Numerical results show that the 3D-COS method is rather correct, fast and competent for pricing the discrete barrier options.
Katok’s entropy formula is an important formula in entropy theory. It plays significant roles in large deviation theories, multifractal analysis, quantitative recurrence and so on. This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems. We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.